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Orthogonal polynomials for the weakly equilibrium Cantor sets


Authors: Gökalp Alpan and Alexander Goncharov
Journal: Proc. Amer. Math. Soc. 144 (2016), 3781-3795
MSC (2010): Primary 42C05, 47B36; Secondary 31A15
DOI: https://doi.org/10.1090/proc/13025
Published electronically: May 6, 2016
MathSciNet review: 3513538
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Abstract: Let $ K(\gamma )$ be the weakly equilibrium Cantor-type set introduced by the second author in an earlier work. It is proven that the monic orthogonal polynomials $ Q_{2^s}$ with respect to the equilibrium measure of $ K(\gamma )$ coincide with the Chebyshev polynomials of the set. Procedures are suggested to find $ Q_{n}$ of all degrees and the corresponding Jacobi parameters. It is shown that the sequence of the Widom factors is bounded below.


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Additional Information

Gökalp Alpan
Affiliation: Department of Mathematics, Bilkent University, 06800 Ankara, Turkey
Email: gokalp@fen.bilkent.edu.tr

Alexander Goncharov
Affiliation: Department of Mathematics, Bilkent University, 06800 Ankara, Turkey
Email: goncha@fen.bilkent.edu.tr

DOI: https://doi.org/10.1090/proc/13025
Keywords: Orthogonal polynomials, equilibrium measure, Cantor sets, Jacobi matrices
Received by editor(s): June 19, 2015
Received by editor(s) in revised form: October 22, 2015
Published electronically: May 6, 2016
Additional Notes: The authors were partially supported by a grant from Tübitak: 115F199.
The authors thank the anonymous referee for pointing out the articles [4, 8, 20–22]
Communicated by: Walter Van Assche
Article copyright: © Copyright 2016 American Mathematical Society

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